The Burgers equation (REF ) with \(\alpha =0\)  and \(\alpha =2\)  has received
an extensive amount of attention since the studies by Burgers in
the 1940s. If \(\alpha =0\) , the equation is perhaps the most basic
example of a PDE evolution leading to shocks; if \(\alpha =2\) , it
provides an accessible model for studying the interaction between
nonlinear and dissipative phenomena. Recently, in [1]} for
the periodic case authors give a complete study for general \(\alpha \in [0,2]\) , see also [2]}, [3]}, [4]}, [5]}.
In particular, for \(\alpha =1\) , with help of the method of modulus of
continuity they proved the global well-posedness of the equation
in the critical Hilbert space \(H^{\frac{1}{2}}(\mathbb {T}^{1})\) .
